Airfoil shape for a turbine rotor blade

ABSTRACT

A turbine rotor blade is provided including an airfoil having an airfoil shape. The turbine rotor blade airfoil having a nominal profile substantially in accordance with Cartesian coordinate values of X, Y and Z set forth in Table I wherein the Cartesian coordinate values of X, Y and Z are non-dimensional values from 0% to 100% convertible to dimensional distances in inches by multiplying the Cartesian coordinate values of X, Y and Z by a height of the airfoil in inches, and wherein X and Y are distances in inches which, when connected by smooth continuing arcs, define airfoil profile sections at each distance Z, the airfoil profile sections at Z distances being joined smoothly with one another to form a complete airfoil shape.

FIELD

The present invention relates to an airfoil for a turbine rotor blade of a stage of a turbine and particularly relates to a shape defining a profile for an airfoil of a turbine rotor blade.

BACKGROUND

Some aircraft and/or power plant systems, for example certain jet aircraft, nuclear, simple cycle and combined cycle power plant systems, employ turbines (also referred to as turbomachines) in their design and operation. Some of these turbines employ airfoils (e.g., turbine rotor blades, blades, airfoils, etc.) which during operation are exposed to fluid flows. These airfoils are configured to aerodynamically interact with the fluid flows and generate energy (e.g., creating thrust, turning kinetic energy to mechanical energy, thermal energy to mechanical energy, etc.) from these fluid flows as part of power generation. As a result of this interaction and conversion, the aerodynamic characteristics and losses of these airfoils have an impact on system and turbine operation, performance, thrust, efficiency, and power.

BRIEF DESCRIPTION

Aspects and advantages are set forth below in the following description, or may be obvious from the description, or may be learned through practice.

An aspect of the present invention may be embodied by a turbine rotor blade including an airfoil having an airfoil shape, the airfoil having a nominal profile substantially in accordance with Cartesian coordinate values of X, Y and Z set forth in Table I wherein the Cartesian coordinate values of X, Y and Z are non-dimensional values from 0% to 100% convertible to dimensional distances in inches by multiplying the Cartesian coordinate values of X, Y and Z by a height of the airfoil in inches, and wherein X and Y are distances in inches which, when connected by smooth continuing arcs, define airfoil profile sections at each distance Z, the airfoil profile sections at Z distances being joined smoothly with one another to form a complete airfoil shape.

An aspect of the present invention may be embodied in a turbine rotor blade including an airfoil having a suction-side uncoated nominal airfoil profile substantially in accordance with suction-side Cartesian coordinate values of X, Y and Z set forth in Table I wherein the Cartesian coordinate values of X, Y and Z are non-dimensional values from 0% to 100% convertible to dimensional distances in inches by multiplying the Cartesian coordinate values of X, Y and Z by a height of the airfoil in inches, and wherein X and Y are distances in inches which, when connected by smooth continuing arcs, define airfoil profile sections at each Z distance, the airfoil profile sections at the Z distances being joined smoothly with one another to form a complete suction-side airfoil shape, the X, Y and Z distances being scalable as a function of the same constant or number to provide a scaled-up or scaled-down airfoil.

An aspect of the present invention may be embodied in a turbine comprising a turbine wheel having a plurality of turbine rotor blades, each of the turbine rotor blades including an airfoil having a suction-side airfoil shape, the airfoil having a nominal profile substantially in accordance with suction-side Cartesian coordinate values of X, Y and Z set forth in Table I wherein the Cartesian coordinate values of X, Y and Z are non-dimensional values from 0% to 100% convertible to dimensional distances by multiplying the Cartesian coordinate values of X, Y and Z by a height of the airfoil, and wherein X and Y are distances which, when connected by smooth continuing arcs, define airfoil profile sections at each Z distance, the airfoil profile sections at the Z distances being joined smoothly with one another to form a complete suction-side airfoil shape.

Those of ordinary skill in the art will better appreciate the features and aspects of such embodiments, and others, upon review of the specification.

BRIEF DESCRIPTION OF THE DRAWINGS

A full and enabling disclosure of various embodiments, including the best mode thereof to one skilled in the art, is set forth more particularly in the remainder of the specification, including reference to the accompanying figures, in which:

FIG. 1 is a schematic representation of an exemplary turbine having multiple stages with each stage including alternating rows of turbine rotor blades and stationary vanes or nozzles according to at least one embodiment of the present disclosure; and

FIG. 2 is a perspective view of a turbine rotor blade according to at least one embodiment of the present disclosure.

DETAILED DESCRIPTION

Referring now to the drawings, particularly to FIG. 1, there is illustrated an exemplary turbine 10 of a gas turbine 12 including a plurality of turbine stages arranged in serial flow order. Each stage of the turbine includes a row of turbine nozzles or vanes disposed axially adjacent to a corresponding row of turbine rotor blades. Four stages are illustrated in FIG. 1. The exact number of stages of the turbine 10 is a choice of engineering design, and may be more or less than the four stages illustrated in FIG. 1. The four stages are merely exemplary of one turbine design, and are not intended to limit the presently claimed turbine rotor blade in any manner.

The first stage comprises a plurality of turbine nozzles or vanes 100 and a plurality of turbine rotor blades 150. The turbine nozzles 100 are annularly arranged about an axis of a turbine rotor 14. The turbine rotor blades 150 are annularly arranged about and coupled to the turbine rotor 14. A second stage of the turbine 12 includes a plurality of turbine nozzles 200 annularly arranged about the axis of the turbine rotor 14 and a plurality of turbine rotor blades 250 annularly arranged about and coupled to the turbine rotor 14. The turbine nozzles 200 may be coupled to a casing 16 of the turbine 12.

As shown in FIG. 1, a third stage of the turbine 12 includes a plurality of turbine nozzles 300 annularly arranged about the axis of the turbine rotor 14 and a plurality of turbine rotor blades 350 annularly arranged about and coupled to the turbine rotor 14. The turbine nozzles 300 may be coupled to the casing 16 of the turbine 12. In particular embodiments, the turbine 12 includes a fourth stage which includes plurality of turbine nozzles 400 annularly arranged about the axis of the turbine rotor 14 and a plurality of turbine rotor blades 450 annularly arranged about and coupled to the turbine rotor 14. The fourth stage turbine nozzles 400 may be coupled to the casing 16 of the turbine 12.

It will be appreciated that the turbine nozzles 100, 200, 300 and 400 and turbine rotor blades 150, 250, 350 and 450 are disposed or at least partially disposed within a hot gas path 18 of the turbine 12. The various stages of the turbine 10 at least partially define the hot gas path 18 through which combustion gases, as indicated by arrows 20, flow during operation of the gas turbine 12.

FIG. 2 provides a perspective view of an exemplary turbine rotor blade 50 as may be incorporated in any stage of the turbine 12. For example, in one embodiment the turbine rotor blade 50 is representative of a turbine rotor blade 150 of the plurality of turbine rotor blades 150 of the first stage. As shown in FIG. 2, the turbine rotor blade 50 includes an airfoil 52 having a pressure-side wall 54 and an opposing suction-side wall 56. The pressure-side wall 54 and the suction-side wall 56 meet or intersect at a leading edge 58 and a trailing edge 60 of the airfoil 52. A chord line 62 extends between the leading edge 58 and the trailing edge 60 such that pressure and suction-side walls 54, 56 can be said to extend in chord or chordwise between the leading edge 58 and the trailing edge 60.

The airfoil 52 includes a root or first end 64 which intersects with and extends radially outwardly from a base or platform 66 of the turbine rotor blade 50. The airfoil 52 terminates radially at a second end or radial tip 68 of the airfoil 52. The pressure and suction-side walls 54, 56 can be said to extend in span or in a span-wise direction 70 between the root 64 and/or the platform 66 and the radial tip 68 of the airfoil 52. In other words, each turbine rotor blade 50 includes an airfoil 52 having opposing pressure and suction-side walls 54, 56 that extend in chord or chordwise 62 between opposing leading and trailing edges 58, 60 and that extend in span or span-wise 70 between the root 64 and the radial tip 68 of the airfoil 52.

In particular configurations, the airfoil 52 may include a fillet 72 formed between the platform 66 and the airfoil 52 proximate to the root 64. The fillet 72 can include a weld or braze fillet, which can be formed via conventional MIG welding, TIG welding, brazing, etc., and can include a profile that can reduce fluid dynamic losses as a result of the presence of fillet 72. In particular embodiments, the platform 66, airfoil 52 and the fillet 72 can be formed as a single component, such as by casting and/or machining and/or 3D printing and/or any other suitable technique now known or later developed and/or discovered. In particular configurations, the turbine rotor blade 50 includes a mounting portion 74 which is formed to connect and/or to secure the turbine rotor blade 50 to the rotor shaft 14.

The airfoil 52 of the turbine rotor blade 50 has a profile at any cross-section taken between the platform 66 or the root 64 and the radial tip 68. The X, Y, and Z values are given in Table I as percentage values of the airfoil length or height/span. As one example only, the height or radial span of the airfoil 52 of turbine rotor blade 50 may be from about 5 inches to about 20 inches. However, it is to be understood that heights below or above this range may also be employed as desired in the specific application.

A gas turbine hot gas path requires airfoils that meet system requirements of aerodynamic and mechanical blade loading and efficiency. To define the airfoil shape of each turbine rotor blade airfoil, there is a unique set or loci of points in space that meet the stage requirements and that can be manufactured. This unique loci of points meets the requirements for stage efficiency and are arrived at by iteration between aerodynamic and mechanical loadings enabling the turbine to run in an efficient, safe and smooth manner. These points are unique and specific to the system and are not obvious to those skilled in the art.

The loci which defines the airfoil profile of turbine rotor blade 50 comprises a set of about 2,200 points with X, Y and Z dimensions relative to a reference origin coordinate system. The Cartesian coordinate system of X, Y and Z values given in Table I below defines the airfoil profile of the turbine rotor blade 50 at various locations along its length and height or radial span. Table I lists data for a non-coated airfoil, and the envelope/tolerance for the coordinates is about +/−5% in a direction normal to any airfoil surface location. A point data origin 76 is defined at or proximate to the root 64 at the leading edge 58. In one embodiment as presented in Table I below, the point data origin 76 is defined at or proximate to (above or below) a transition or intersection line 78 defined between the fillet 72 and the airfoil 52. The point data origin 76 corresponds to the non-dimensional Z value of Table I (presented below) at Z equals 0.

The coordinate values for the X, Y and Z coordinates are set forth in non-dimensionalized units by the airfoil height in Table I although other units of dimensions may be used when the values are appropriately converted. The X, Y, and Z values set forth in Table I are also expressed in non-dimensional form (X, Y, and Z) from 0% to 100% of the airfoil 52 height or radial span 70. To convert the Z value to a Z coordinate value, e.g., in inches, the non-dimensional Z value given in Table I is multiplied by the Z length of the airfoil in inches. As described above, the Cartesian coordinate system has orthogonally-related X, Y and Z axes and the X axis lies generally parallel to a centerline of the rotor shaft 14, i.e., the rotary axis and a positive X coordinate value is axial toward an aft, i.e., exhaust end of the turbine 10. The positive Y coordinate value extends tangentially in the direction of rotation of the rotor and the positive Z coordinate value is radially outwardly toward the radial tip 68. All the values in Table I are given at room temperature and do not include the fillet 72.

By defining X and Y coordinate values at selected locations in a Z direction normal to the X, Y plane, the profile section or airfoil shape of the airfoil 52 of the turbine rotor blade 50, at each Z distance along the length of the airfoil 52 can be ascertained. By connecting the X and Y values with smooth continuing arcs, each profile section at each distance Z is fixed. The airfoil profiles of the various surface locations between the distances Z are determined by smoothly connecting the adjacent profile sections to one another to form the airfoil profile.

The Table I values are generated and shown to four decimal places for determining the profile of the airfoil 52. As the turbine rotor blade 50 heats up during operation of the gas turbine 12, surface stress and temperature will cause a change in the X, Y and Z's. Accordingly, the values for the profile given in Table I represent ambient, non-operating or non-hot conditions (e.g., room temperature) and are for an uncoated airfoil.

There are typical manufacturing tolerances as well as coatings which must be accounted for in the actual profile of the airfoil 52. Each cross-section is joined smoothly with the other cross-sections to form the complete airfoil shape. It will therefore be appreciated that +/− typical manufacturing tolerances, i.e., +/− values, including any coating thicknesses, are additive to the X and Y values given in Table I below. Accordingly, a distance of +/−5% in a direction normal to any surface location along the airfoil profile defines an airfoil profile envelope for this particular turbine rotor blade 50 airfoil design and turbine 12, i.e., a range of variation between measured points on the actual airfoil surface at nominal cold or room temperature and the ideal position of those points as given in Table I below at the same temperature. The data provided in Table I is scalable and the geometry pertains to all aerodynamic scales, at, above and/or below a 3000-3600 RPM range. The design of the airfoil 52 for turbine rotor blade 50 is robust to this range of variation without impairment of mechanical and aerodynamic functions.

TABLE I SUCTION-SIDE PRESSURE-SIDE X (%) Y (%) Z (%) N X (%) Y (%) Z (%) 0.0000 0.0000 0.0000 1 0.0000 0.0000 0.0000 −0.6179 0.9061 0.0000 2 0.6206 −0.3865 0.0000 −0.8829 1.9769 0.0000 3 1.3301 −0.5682 0.0000 −0.9152 3.0815 0.0000 4 2.0634 −0.5851 0.0000 −0.7952 4.1807 0.0000 5 2.7924 −0.4976 0.0000 −0.5743 5.2645 0.0000 6 3.5131 −0.3556 0.0000 −0.2815 6.3313 0.0000 7 4.2279 −0.1862 0.0000 0.0680 7.3809 0.0000 8 4.9385 −0.0002 0.0000 0.4649 8.4135 0.0000 9 5.6456 0.1989 0.0000 0.9019 9.4299 0.0000 10 6.3495 0.4091 0.0000 1.3731 10.4309 0.0000 11 7.0506 0.6284 0.0000 1.8738 11.4174 0.0000 12 7.7493 0.8551 0.0000 2.4007 12.3902 0.0000 13 8.4460 1.0878 0.0000 2.9513 13.3498 0.0000 14 9.1412 1.3251 0.0000 3.5239 14.2965 0.0000 15 9.8353 1.5657 0.0000 4.1176 15.2300 0.0000 16 10.5287 1.8083 0.0000 4.7319 16.1502 0.0000 17 11.2217 2.0519 0.0000 5.3669 17.0561 0.0000 18 11.9149 2.2951 0.0000 6.0231 17.9469 0.0000 19 12.6085 2.5370 0.0000 6.7007 18.8214 0.0000 20 13.3030 2.7764 0.0000 7.4006 19.6782 0.0000 21 13.9987 3.0122 0.0000 8.1232 20.5160 0.0000 22 14.6959 3.2434 0.0000 8.8692 21.3330 0.0000 23 15.3950 3.4690 0.0000 9.6390 22.1275 0.0000 24 16.0962 3.6879 0.0000 10.4333 22.8977 0.0000 25 16.7998 3.8992 0.0000 11.2524 23.6413 0.0000 26 17.5059 4.1018 0.0000 12.0968 24.3561 0.0000 27 18.2147 4.2947 0.0000 12.9669 25.0393 0.0000 28 18.9263 4.4769 0.0000 13.8630 25.6881 0.0000 29 19.6409 4.6473 0.0000 14.7850 26.2994 0.0000 30 20.3583 4.8050 0.0000 15.7329 26.8699 0.0000 31 21.0786 4.9489 0.0000 16.7060 27.3960 0.0000 32 21.8018 5.0780 0.0000 17.7035 27.8743 0.0000 33 22.5276 5.1912 0.0000 18.7242 28.3010 0.0000 34 23.2558 5.2874 0.0000 19.7662 28.6724 0.0000 35 23.9862 5.3658 0.0000 20.8273 28.9850 0.0000 36 24.7184 5.4254 0.0000 21.9048 29.2354 0.0000 37 25.4519 5.4653 0.0000 22.9954 29.4204 0.0000 38 26.1862 5.4848 0.0000 24.0954 29.5377 0.0000 39 26.9207 5.4830 0.0000 25.2005 29.5860 0.0000 40 27.6549 5.4596 0.0000 26.3065 29.5649 0.0000 41 28.3880 5.4138 0.0000 27.4090 29.4749 0.0000 42 29.1194 5.3454 0.0000 28.5039 29.3169 0.0000 43 29.8483 5.2542 0.0000 29.5871 29.0926 0.0000 44 30.5739 5.1399 0.0000 30.6551 28.8042 0.0000 45 31.2955 5.0026 0.0000 31.7047 28.4548 0.0000 46 32.0124 4.8425 0.0000 32.7333 28.0478 0.0000 47 32.7239 4.6597 0.0000 33.7391 27.5873 0.0000 48 33.4292 4.4547 0.0000 34.7207 27.0770 0.0000 49 34.1279 4.2278 0.0000 35.6768 26.5204 0.0000 50 34.8193 3.9796 0.0000 36.6065 25.9208 0.0000 51 35.5029 3.7108 0.0000 37.5090 25.2811 0.0000 52 36.1783 3.4220 0.0000 38.3839 24.6040 0.0000 53 36.8451 3.1139 0.0000 39.2307 23.8920 0.0000 54 37.5031 2.7873 0.0000 40.0493 23.1479 0.0000 55 38.1519 2.4430 0.0000 40.8400 22.3741 0.0000 56 38.7915 2.0817 0.0000 41.6029 21.5729 0.0000 57 39.4217 1.7042 0.0000 42.3388 20.7468 0.0000 58 40.0424 1.3113 0.0000 43.0483 19.8979 0.0000 59 40.6536 0.9038 0.0000 43.7324 19.0285 0.0000 60 41.2553 0.4825 0.0000 44.3922 18.1405 0.0000 61 41.8475 0.0479 0.0000 45.0290 17.2358 0.0000 62 42.4304 −0.3991 0.0000 45.6442 16.3162 0.0000 63 43.0040 −0.8580 0.0000 46.2391 15.3834 0.0000 64 43.5685 −1.3281 0.0000 46.8151 14.4389 0.0000 65 44.1239 −1.8088 0.0000 47.3736 13.4838 0.0000 66 44.6704 −2.2997 0.0000 47.9159 12.5195 0.0000 67 45.2081 −2.8002 0.0000 48.4431 11.5468 0.0000 68 45.7372 −3.3098 0.0000 48.9562 10.5666 0.0000 69 46.2577 −3.8281 0.0000 49.4560 9.5796 0.0000 70 46.7698 −4.3548 0.0000 49.9434 8.5864 0.0000 71 47.2735 −4.8894 0.0000 50.4191 7.5875 0.0000 72 47.7690 −5.4318 0.0000 50.8837 6.5834 0.0000 73 48.2563 −5.9814 0.0000 51.3377 5.5745 0.0000 74 48.7357 −6.5381 0.0000 51.7816 4.5611 0.0000 75 49.2071 −7.1014 0.0000 52.2159 3.5435 0.0000 76 49.6710 −7.6710 0.0000 52.6410 2.5221 0.0000 77 50.1274 −8.2466 0.0000 53.0574 1.4970 0.0000 78 50.5766 −8.8278 0.0000 53.4654 0.4687 0.0000 79 51.0189 −9.4143 0.0000 53.8656 −0.5628 0.0000 80 51.4546 −10.0057 0.0000 54.2585 −1.5970 0.0000 81 51.8838 −10.6019 0.0000 54.6447 −2.6338 0.0000 82 52.3068 −11.2025 0.0000 55.0246 −3.6729 0.0000 83 52.7237 −11.8073 0.0000 55.3990 −4.7140 0.0000 84 53.1346 −12.4162 0.0000 55.7684 −5.7569 0.0000 85 53.5398 −13.0289 0.0000 56.1332 −6.8014 0.0000 86 53.9393 −13.6454 0.0000 56.4939 −7.8473 0.0000 87 54.3333 −14.2654 0.0000 56.8505 −8.8946 0.0000 88 54.7219 −14.8888 0.0000 57.2033 −9.9432 0.0000 89 55.1051 −15.5155 0.0000 57.5523 −10.9931 0.0000 90 55.4831 −16.1454 0.0000 57.8974 −12.0443 0.0000 91 55.8558 −16.7784 0.0000 58.2386 −13.0967 0.0000 92 56.2234 −17.4144 0.0000 58.5756 −14.1505 0.0000 93 56.5859 −18.0533 0.0000 58.9084 −15.2057 0.0000 94 56.9434 −18.6950 0.0000 59.2367 −16.2622 0.0000 95 57.2961 −19.3394 0.0000 59.5605 −17.3201 0.0000 96 57.6440 −19.9864 0.0000 59.8795 −18.3795 0.0000 97 57.9873 −20.6358 0.0000 60.1937 −19.4403 0.0000 98 58.3934 −21.2417 0.0000 60.3912 −20.5182 0.0000 99 59.0573 −21.5273 0.0000 59.7670 −21.3847 0.0000 100 59.7670 −21.3847 0.0000 0.3825 0.6250 10.0000 1 0.3825 0.6250 10.0000 −0.0778 1.6429 10.0000 2 0.9061 0.1018 10.0000 −0.2488 2.7510 10.0000 3 1.5774 −0.2104 10.0000 −0.2388 3.8733 10.0000 4 2.3117 −0.3153 10.0000 −0.1077 4.9884 10.0000 5 3.0526 −0.2593 10.0000 0.1107 6.0899 10.0000 6 3.7837 −0.1235 10.0000 0.3963 7.1760 10.0000 7 4.5074 0.0485 10.0000 0.7360 8.2464 10.0000 8 5.2263 0.2394 10.0000 1.1207 9.3015 10.0000 9 5.9420 0.4417 10.0000 1.5441 10.3418 10.0000 10 6.6551 0.6533 10.0000 2.0017 11.3674 10.0000 11 7.3655 0.8739 10.0000 2.4901 12.3788 10.0000 12 8.0732 1.1028 10.0000 3.0065 13.3761 10.0000 13 8.7784 1.3392 10.0000 3.5487 14.3597 10.0000 14 9.4817 1.5816 10.0000 4.1148 15.3297 10.0000 15 10.1832 1.8287 10.0000 4.7034 16.2862 10.0000 16 10.8837 2.0789 10.0000 5.3140 17.2289 10.0000 17 11.5836 2.3306 10.0000 5.9467 18.1568 10.0000 18 12.2835 2.5827 10.0000 6.6019 19.0690 10.0000 19 12.9836 2.8337 10.0000 7.2802 19.9641 10.0000 20 13.6846 3.0826 10.0000 7.9824 20.8407 10.0000 21 14.3866 3.3283 10.0000 8.7091 21.6970 10.0000 22 15.0901 3.5700 10.0000 9.4610 22.5313 10.0000 23 15.7953 3.8065 10.0000 10.2387 23.3415 10.0000 24 16.5026 4.0369 10.0000 11.0428 24.1255 10.0000 25 17.2122 4.2599 10.0000 11.8740 24.8808 10.0000 26 17.9244 4.4743 10.0000 12.7327 25.6046 10.0000 27 18.6396 4.6787 10.0000 13.6193 26.2940 10.0000 28 19.3579 4.8717 10.0000 14.5340 26.9457 10.0000 29 20.0795 5.0523 10.0000 15.4766 27.5562 10.0000 30 20.8043 5.2191 10.0000 16.4469 28.1217 10.0000 31 21.5324 5.3714 10.0000 17.4441 28.6381 10.0000 32 22.2636 5.5080 10.0000 18.4672 29.1011 10.0000 33 22.9977 5.6277 10.0000 19.5144 29.5066 10.0000 34 23.7345 5.7291 10.0000 20.5835 29.8505 10.0000 35 24.4738 5.8110 10.0000 21.6715 30.1285 10.0000 36 25.2150 5.8724 10.0000 22.7749 30.3370 10.0000 37 25.9578 5.9122 10.0000 23.8895 30.4738 10.0000 38 26.7014 5.9296 10.0000 25.0106 30.5367 10.0000 39 27.4451 5.9240 10.0000 26.1335 30.5238 10.0000 40 28.1883 5.8944 10.0000 27.2530 30.4357 10.0000 41 28.9301 5.8403 10.0000 28.3642 30.2742 10.0000 42 29.6697 5.7614 10.0000 29.4627 30.0408 10.0000 43 30.4062 5.6575 10.0000 30.5440 29.7380 10.0000 44 31.1387 5.5286 10.0000 31.6050 29.3699 10.0000 45 31.8664 5.3745 10.0000 32.6428 28.9410 10.0000 46 32.5883 5.1956 10.0000 33.6553 28.4552 10.0000 47 33.3038 4.9922 10.0000 34.6408 27.9167 10.0000 48 34.0120 4.7648 10.0000 35.5984 27.3299 10.0000 49 34.7122 4.5140 10.0000 36.5275 26.6991 10.0000 50 35.4040 4.2407 10.0000 37.4281 26.0281 10.0000 51 36.0867 3.9457 10.0000 38.3003 25.3206 10.0000 52 36.7601 3.6297 10.0000 39.1442 24.5796 10.0000 53 37.4237 3.2938 10.0000 39.9606 23.8083 10.0000 54 38.0773 2.9388 10.0000 40.7498 23.0093 10.0000 55 38.7207 2.5655 10.0000 41.5127 22.1850 10.0000 56 39.3537 2.1750 10.0000 42.2499 21.3378 10.0000 57 39.9763 1.7680 10.0000 42.9621 20.4694 10.0000 58 40.5884 1.3456 10.0000 43.6502 19.5817 10.0000 59 41.1903 0.9085 10.0000 44.3150 18.6765 10.0000 60 41.7818 0.4576 10.0000 44.9574 17.7553 10.0000 61 42.3633 −0.0063 10.0000 45.5784 16.8195 10.0000 62 42.9348 −0.4823 10.0000 46.1792 15.8705 10.0000 63 43.4965 −0.9699 10.0000 46.7606 14.9096 10.0000 64 44.0487 −1.4683 10.0000 47.3236 13.9378 10.0000 65 44.5914 −1.9769 10.0000 47.8694 12.9562 10.0000 66 45.1249 −2.4952 10.0000 48.3989 11.9657 10.0000 67 45.6494 −3.0226 10.0000 48.9130 10.9671 10.0000 68 46.1650 −3.5587 10.0000 49.4126 9.9612 10.0000 69 46.6718 −4.1031 10.0000 49.8987 8.9487 10.0000 70 47.1700 −4.6554 10.0000 50.3721 7.9302 10.0000 71 47.6598 −5.2153 10.0000 50.8334 6.9062 10.0000 72 48.1411 −5.7824 10.0000 51.2836 5.8772 10.0000 73 48.6141 −6.3564 10.0000 51.7232 4.8437 10.0000 74 49.0790 −6.9370 10.0000 52.1529 3.8060 10.0000 75 49.5360 −7.5239 10.0000 52.5733 2.7645 10.0000 76 49.9852 −8.1168 10.0000 52.9850 1.7196 10.0000 77 50.4269 −8.7153 10.0000 53.3885 0.6714 10.0000 78 50.8614 −9.3190 10.0000 53.7841 −0.3798 10.0000 79 51.2891 −9.9276 10.0000 54.1720 −1.4338 10.0000 80 51.7100 −10.5408 10.0000 54.5525 −2.4905 10.0000 81 52.1246 −11.1584 10.0000 54.9259 −3.5498 10.0000 82 52.5328 −11.7802 10.0000 55.2927 −4.6113 10.0000 83 52.9349 −12.4059 10.0000 55.6539 −5.6748 10.0000 84 53.3309 −13.0355 10.0000 56.0103 −6.7399 10.0000 85 53.7211 −13.6688 10.0000 56.3629 −7.8062 10.0000 86 54.1054 −14.3056 10.0000 56.7123 −8.8737 10.0000 87 54.4840 −14.9459 10.0000 57.0586 −9.9421 10.0000 88 54.8570 −15.5894 10.0000 57.4017 −11.0116 10.0000 89 55.2246 −16.2361 10.0000 57.7415 −12.0821 10.0000 90 55.5866 −16.8858 10.0000 58.0776 −13.1537 10.0000 91 55.9434 −17.5385 10.0000 58.4097 −14.2267 10.0000 92 56.2949 −18.1941 10.0000 58.7377 −15.3009 10.0000 93 56.6411 −18.8524 10.0000 59.0614 −16.3763 10.0000 94 56.9822 −19.5134 10.0000 59.3808 −17.4531 10.0000 95 57.3182 −20.1770 10.0000 59.6962 −18.5311 10.0000 96 57.6490 −20.8432 10.0000 60.0073 −19.6103 10.0000 97 57.9748 −21.5119 10.0000 60.3140 −20.6907 10.0000 98 58.3209 −22.1687 10.0000 60.4217 −21.7880 10.0000 99 58.9267 −22.5779 10.0000 59.6588 −22.5572 10.0000 100 59.6588 −22.5572 10.0000 1.4533 1.2803 20.0000 1 1.4533 1.2803 20.0000 0.9283 2.2732 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11.5869 25.5671 80.0000 16 16.3148 12.5681 80.0000 12.3502 26.2679 80.0000 17 17.0502 12.7460 80.0000 13.1459 26.9317 80.0000 18 17.7909 12.9003 80.0000 13.9736 27.5553 80.0000 19 18.5363 13.0302 80.0000 14.8323 28.1353 80.0000 20 19.2856 13.1347 80.0000 15.7207 28.6686 80.0000 21 20.0381 13.2129 80.0000 16.6373 29.1519 80.0000 22 20.7930 13.2632 80.0000 17.5802 29.5817 80.0000 23 21.5493 13.2843 80.0000 18.5470 29.9544 80.0000 24 22.3058 13.2751 80.0000 19.5350 30.2668 80.0000 25 23.0612 13.2342 80.0000 20.5408 30.5157 80.0000 26 23.8142 13.1611 80.0000 21.5607 30.6989 80.0000 27 24.5633 13.0550 80.0000 22.5903 30.8149 80.0000 28 25.3070 12.9158 80.0000 23.6254 30.8630 80.0000 29 26.0437 12.7436 80.0000 24.6614 30.8432 80.0000 30 26.7720 12.5388 80.0000 25.6939 30.7564 80.0000 31 27.4905 12.3020 80.0000 26.7188 30.6039 80.0000 32 28.1981 12.0342 80.0000 27.7321 30.3876 80.0000 33 28.8937 11.7367 80.0000 28.7304 30.1099 80.0000 34 29.5766 11.4110 80.0000 29.7106 29.7737 80.0000 35 30.2462 11.0588 80.0000 30.6699 29.3820 80.0000 36 30.9021 10.6819 80.0000 31.6063 28.9383 80.0000 37 31.5444 10.2820 80.0000 32.5181 28.4458 80.0000 38 32.1731 9.8611 80.0000 33.4039 27.9081 80.0000 39 32.7883 9.4207 80.0000 34.2628 27.3284 80.0000 40 33.3903 8.9625 80.0000 35.0942 26.7099 80.0000 41 33.9796 8.4879 80.0000 35.8978 26.0556 80.0000 42 34.5564 7.9983 80.0000 36.6733 25.3682 80.0000 43 35.1212 7.4949 80.0000 37.4205 24.6502 80.0000 44 35.6746 6.9789 80.0000 38.1393 23.9036 80.0000 45 36.2169 6.4513 80.0000 38.8293 23.1305 80.0000 46 36.7487 5.9131 80.0000 39.4909 22.3328 80.0000 47 37.2705 5.3653 80.0000 40.1252 21.5133 80.0000 48 37.7829 4.8085 80.0000 40.7344 20.6750 80.0000 49 38.2863 4.2437 80.0000 41.3210 19.8207 80.0000 50 38.7813 3.6715 80.0000 41.8874 18.9528 80.0000 51 39.2683 3.0924 80.0000 42.4357 18.0734 80.0000 52 39.7478 2.5072 80.0000 42.9678 17.1841 80.0000 53 40.2204 1.9163 80.0000 43.4854 16.2863 80.0000 54 40.6863 1.3202 80.0000 43.9899 15.3810 80.0000 55 41.1460 0.7193 80.0000 44.4825 14.4692 80.0000 56 41.5999 0.1139 80.0000 44.9643 13.5517 80.0000 57 42.0482 −0.4956 80.0000 45.4360 12.6289 80.0000 58 42.4913 −1.1089 80.0000 45.8985 11.7015 80.0000 59 42.9294 −1.7258 80.0000 46.3523 10.7698 80.0000 60 43.3627 −2.3460 80.0000 46.7978 9.8341 80.0000 61 43.7916 −2.9693 80.0000 47.2356 8.8948 80.0000 62 44.2162 −3.5956 80.0000 47.6661 7.9521 80.0000 63 44.6366 −4.2246 80.0000 48.0895 7.0062 80.0000 64 45.0532 −4.8562 80.0000 48.5063 6.0573 80.0000 65 45.4659 −5.4904 80.0000 48.9166 5.1056 80.0000 66 45.8750 −6.1269 80.0000 49.3207 4.1513 80.0000 67 46.2805 −6.7656 80.0000 49.7189 3.1945 80.0000 68 46.6827 −7.4065 80.0000 50.1112 2.2353 80.0000 69 47.0815 −8.0494 80.0000 50.4979 1.2738 80.0000 70 47.4772 −8.6943 80.0000 50.8791 0.3101 80.0000 71 47.8699 −9.3411 80.0000 51.2551 −0.6556 80.0000 72 48.2597 −9.9896 80.0000 51.6257 −1.6234 80.0000 73 48.6466 −10.6398 80.0000 51.9913 −2.5931 80.0000 74 49.0308 −11.2916 80.0000 52.3520 −3.5647 80.0000 75 49.4124 −11.9449 80.0000 52.7078 −4.5381 80.0000 76 49.7915 −12.5997 80.0000 53.0590 −5.5131 80.0000 77 50.1683 −13.2558 80.0000 53.4057 −6.4897 80.0000 78 50.5427 −13.9133 80.0000 53.7480 −7.4679 80.0000 79 50.9150 −14.5720 80.0000 54.0863 −8.4475 80.0000 80 51.2852 −15.2318 80.0000 54.4206 −9.4284 80.0000 81 51.6535 −15.8928 80.0000 54.7514 −10.4106 80.0000 82 52.0198 −16.5548 80.0000 55.0788 −11.3939 80.0000 83 52.3844 −17.2178 80.0000 55.4032 −12.3781 80.0000 84 52.7472 −17.8817 80.0000 55.7247 −13.3634 80.0000 85 53.1085 −18.5465 80.0000 56.0435 −14.3495 80.0000 86 53.4682 −19.2122 80.0000 56.3597 −15.3364 80.0000 87 53.8264 −19.8786 80.0000 56.6734 −16.3241 80.0000 88 54.1833 −20.5458 80.0000 56.9844 −17.3127 80.0000 89 54.5389 −21.2136 80.0000 57.2926 −18.3022 80.0000 90 54.8932 −21.8822 80.0000 57.5979 −19.2925 80.0000 91 55.2463 −22.5513 80.0000 57.8999 −20.2839 80.0000 92 55.5984 −23.2210 80.0000 58.1983 −21.2764 80.0000 93 55.9494 −23.8913 80.0000 58.4928 −22.2700 80.0000 94 56.2994 −24.5621 80.0000 58.7833 −23.2648 80.0000 95 56.6485 −25.2333 80.0000 59.0694 −24.2609 80.0000 96 56.9967 −25.9051 80.0000 59.3511 −25.2582 80.0000 97 57.3441 −26.5772 80.0000 59.6282 −26.2568 80.0000 98 57.6906 −27.2498 80.0000 59.6903 −27.2679 80.0000 99 58.1846 −27.8045 80.0000 58.9193 −27.8992 80.0000 100 58.9193 −27.8992 80.0000 5.9164 15.4629 90.0000 1 5.9164 15.4629 90.0000 5.6740 16.4324 90.0000 2 6.2777 14.8104 90.0000 5.6656 17.4331 90.0000 3 6.7828 14.2616 90.0000 5.8321 18.4206 90.0000 4 7.3953 13.8352 90.0000 6.1264 19.3783 90.0000 5 8.0753 13.5266 90.0000 6.5146 20.3022 90.0000 6 8.7933 13.3204 90.0000 6.9746 21.1926 90.0000 7 9.5304 13.1978 90.0000 7.4929 22.0504 90.0000 8 10.2758 13.1426 90.0000 8.0602 22.8767 90.0000 9 11.0232 13.1447 90.0000 8.6691 23.6729 90.0000 10 11.7688 13.1971 90.0000 9.3141 24.4401 90.0000 11 12.5106 13.2900 90.0000 9.9908 25.1796 90.0000 12 13.2487 13.4085 90.0000 10.6962 25.8918 90.0000 13 13.9851 13.5371 90.0000 11.4287 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12.2169 90.0000 29.9498 30.3864 90.0000 34 29.3013 11.8913 90.0000 30.8570 29.9604 90.0000 35 29.9624 11.5423 90.0000 31.7390 29.4841 90.0000 36 30.6112 11.1711 90.0000 32.5946 28.9621 90.0000 37 31.2476 10.7788 90.0000 33.4234 28.3983 90.0000 38 31.8714 10.3668 90.0000 34.2250 27.7966 90.0000 39 32.4825 9.9363 90.0000 34.9994 27.1602 90.0000 40 33.0813 9.4887 90.0000 35.7463 26.4917 90.0000 41 33.6679 9.0253 90.0000 36.4658 25.7938 90.0000 42 34.2426 8.5472 90.0000 37.1582 25.0690 90.0000 43 34.8058 8.0556 90.0000 37.8243 24.3200 90.0000 44 35.3580 7.5517 90.0000 38.4653 23.5493 90.0000 45 35.8995 7.0363 90.0000 39.0825 22.7595 90.0000 46 36.4308 6.5104 90.0000 39.6777 21.9529 90.0000 47 36.9522 5.9746 90.0000 40.2528 21.1319 90.0000 48 37.4641 5.4298 90.0000 40.8094 20.2982 90.0000 49 37.9670 4.8767 90.0000 41.3493 19.4537 90.0000 50 38.4612 4.3157 90.0000 41.8740 18.5995 90.0000 51 38.9471 3.7476 90.0000 42.3846 17.7369 90.0000 52 39.4250 3.1728 90.0000 42.8823 16.8667 90.0000 53 39.8955 2.5918 90.0000 43.3680 15.9898 90.0000 54 40.3588 2.0051 90.0000 43.8425 15.1068 90.0000 55 40.8153 1.4130 90.0000 44.3066 14.2183 90.0000 56 41.2653 0.8161 90.0000 44.7610 13.3248 90.0000 57 41.7092 0.2146 90.0000 45.2063 12.4267 90.0000 58 42.1474 −0.3911 90.0000 45.6431 11.5245 90.0000 59 42.5801 −1.0008 90.0000 46.0719 10.6184 90.0000 60 43.0076 −1.6141 90.0000 46.4931 9.7088 90.0000 61 43.4302 −2.2308 90.0000 46.9073 8.7959 90.0000 62 43.8481 −2.8506 90.0000 47.3148 7.8800 90.0000 63 44.2615 −3.4735 90.0000 47.7159 6.9614 90.0000 64 44.6707 −4.0991 90.0000 48.1110 6.0401 90.0000 65 45.0758 −4.7275 90.0000 48.5003 5.1163 90.0000 66 45.4769 −5.3583 90.0000 48.8840 4.1902 90.0000 67 45.8743 −5.9915 90.0000 49.2623 3.2620 90.0000 68 46.2681 −6.6270 90.0000 49.6355 2.3316 90.0000 69 46.6583 −7.2647 90.0000 50.0037 1.3992 90.0000 70 47.0451 −7.9044 90.0000 50.3669 0.4649 90.0000 71 47.4287 −8.5461 90.0000 50.7254 −0.4712 90.0000 72 47.8090 −9.1897 90.0000 51.0792 −1.4092 90.0000 73 48.1863 −9.8351 90.0000 51.4284 −2.3488 90.0000 74 48.5606 −10.4822 90.0000 51.7733 −3.2900 90.0000 75 48.9320 −11.1310 90.0000 52.1138 −4.2328 90.0000 76 49.3007 −11.7814 90.0000 52.4501 −5.1772 90.0000 77 49.6667 −12.4332 90.0000 52.7825 −6.1229 90.0000 78 50.0303 −13.0865 90.0000 53.1110 −7.0699 90.0000 79 50.3915 −13.7410 90.0000 53.4360 −8.0182 90.0000 80 50.7504 −14.3967 90.0000 53.7576 −8.9677 90.0000 81 51.1074 −15.0536 90.0000 54.0761 −9.9181 90.0000 82 51.4625 −15.7115 90.0000 54.3919 −10.8695 90.0000 83 51.8158 −16.3703 90.0000 54.7052 −11.8218 90.0000 84 52.1675 −17.0300 90.0000 55.0163 −12.7747 90.0000 85 52.5176 −17.6905 90.0000 55.3254 −13.7283 90.0000 86 52.8665 −18.3517 90.0000 55.6327 −14.6824 90.0000 87 53.2140 −19.0136 90.0000 55.9382 −15.6372 90.0000 88 53.5602 −19.6762 90.0000 56.2418 −16.5925 90.0000 89 53.9054 −20.3394 90.0000 56.5435 −17.5485 90.0000 90 54.2494 −21.0031 90.0000 56.8430 −18.5051 90.0000 91 54.5923 −21.6674 90.0000 57.1400 −19.4625 90.0000 92 54.9343 −22.3322 90.0000 57.4342 −20.4208 90.0000 93 55.2753 −22.9975 90.0000 57.7254 −21.3800 90.0000 94 55.6154 −23.6632 90.0000 58.0132 −22.3402 90.0000 95 55.9547 −24.3293 90.0000 58.2975 −23.3015 90.0000 96 56.2932 −24.9959 90.0000 58.5780 −24.2639 90.0000 97 56.6311 −25.6628 90.0000 58.8546 −25.2274 90.0000 98 56.9684 −26.3299 90.0000 58.9449 −26.2062 90.0000 99 57.5034 −26.8289 90.0000 58.2361 −26.8609 90.0000 100 58.2361 −26.8609 90.0000 4.4830 18.6636 100.0000 1 4.4830 18.6636 100.0000 4.3026 19.6910 100.0000 2 4.8000 17.9716 100.0000 4.3850 20.7320 100.0000 3 5.2582 17.3634 100.0000 4.6674 21.7384 100.0000 4 5.8236 16.8528 100.0000 5.0931 22.6936 100.0000 5 6.4635 16.4389 100.0000 5.6252 23.5942 100.0000 6 7.1523 16.1120 100.0000 6.2387 24.4415 100.0000 7 7.8720 15.8602 100.0000 6.9147 25.2401 100.0000 8 8.6111 15.6722 100.0000 7.6377 25.9965 100.0000 9 9.3619 15.5384 100.0000 8.3958 26.7177 100.0000 10 10.1196 15.4508 100.0000 9.1802 27.4103 100.0000 11 10.8809 15.4039 100.0000 9.9851 28.0791 100.0000 12 11.6436 15.3943 100.0000 10.8071 28.7267 100.0000 13 12.4059 15.4203 100.0000 11.6447 29.3540 100.0000 14 13.1663 15.4798 100.0000 12.4976 29.9603 100.0000 15 13.9238 15.5692 100.0000 13.3663 30.5436 100.0000 16 14.6783 15.6817 100.0000 14.2516 31.1015 100.0000 17 15.4307 15.8073 100.0000 15.1540 31.6313 100.0000 18 16.1829 15.9340 100.0000 16.0737 32.1306 100.0000 19 16.9368 16.0500 100.0000 17.0102 32.5974 100.0000 20 17.6937 16.1449 100.0000 17.9630 33.0299 100.0000 21 18.4535 16.2116 100.0000 18.9313 33.4267 100.0000 22 19.2154 16.2469 100.0000 19.9141 33.7860 100.0000 23 19.9782 16.2502 100.0000 20.9104 34.1059 100.0000 24 20.7405 16.2226 100.0000 21.9191 34.3841 100.0000 25 21.5011 16.1654 100.0000 22.9391 34.6177 100.0000 26 22.2589 16.0793 100.0000 23.9688 34.8036 100.0000 27 23.0130 15.9641 100.0000 25.0064 34.9387 100.0000 28 23.7619 15.8193 100.0000 26.0496 35.0205 100.0000 29 24.5043 15.6445 100.0000 27.0956 35.0470 100.0000 30 25.2390 15.4394 100.0000 28.1415 35.0166 100.0000 31 25.9647 15.2044 100.0000 29.1840 34.9278 100.0000 32 26.6803 14.9405 100.0000 30.2198 34.7793 100.0000 33 27.3852 14.6490 100.0000 31.2450 34.5702 100.0000 34 28.0786 14.3312 100.0000 32.2558 34.3001 100.0000 35 28.7601 13.9886 100.0000 33.2486 33.9696 100.0000 36 29.4293 13.6226 100.0000 34.2197 33.5802 100.0000 37 30.0860 13.2345 100.0000 35.1660 33.1338 100.0000 38 30.7300 12.8258 100.0000 36.0847 32.6329 100.0000 39 31.3612 12.3975 100.0000 36.9731 32.0802 100.0000 40 31.9797 11.9511 100.0000 37.8295 31.4790 100.0000 41 32.5856 11.4877 100.0000 38.6522 30.8325 100.0000 42 33.1791 11.0084 100.0000 39.4403 30.1441 100.0000 43 33.7603 10.5144 100.0000 40.1931 29.4174 100.0000 44 34.3295 10.0067 100.0000 40.9105 28.6557 100.0000 45 34.8871 9.4862 100.0000 41.5930 27.8625 100.0000 46 35.4334 8.9538 100.0000 42.2414 27.0412 100.0000 47 35.9688 8.4104 100.0000 42.8570 26.1951 100.0000 48 36.4936 7.8568 100.0000 43.4417 25.3272 100.0000 49 37.0082 7.2938 100.0000 43.9975 24.4406 100.0000 50 37.5130 6.7219 100.0000 44.5266 23.5377 100.0000 51 38.0085 6.1419 100.0000 45.0311 22.6210 100.0000 52 38.4949 5.5544 100.0000 45.5133 21.6922 100.0000 53 38.9728 4.9598 100.0000 45.9750 20.7531 100.0000 54 39.4425 4.3587 100.0000 46.4180 19.8051 100.0000 55 39.9043 3.7516 100.0000 46.8439 18.8492 100.0000 56 40.3586 3.1388 100.0000 47.2540 17.8864 100.0000 57 40.8058 2.5209 100.0000 47.6496 16.9176 100.0000 58 41.2463 1.8981 100.0000 48.0318 15.9435 100.0000 59 41.6803 1.2708 100.0000 48.4018 14.9646 100.0000 60 42.1082 0.6393 100.0000 48.7604 13.9815 100.0000 61 42.5302 0.0038 100.0000 49.1085 12.9946 100.0000 62 42.9466 −0.6353 100.0000 49.4469 12.0044 100.0000 63 43.3577 −1.2779 100.0000 49.7763 11.0111 100.0000 64 43.7636 −1.9237 100.0000 50.0973 10.0151 100.0000 65 44.1647 −2.5725 100.0000 50.4106 9.0166 100.0000 66 44.5611 −3.2243 100.0000 50.7165 8.0159 100.0000 67 44.9530 −3.8787 100.0000 51.0156 7.0131 100.0000 68 45.3405 −4.5357 100.0000 51.3083 6.0083 100.0000 69 45.7240 −5.1952 100.0000 51.5948 5.0018 100.0000 70 46.1034 −5.8569 100.0000 51.8754 3.9937 100.0000 71 46.4790 −6.5209 100.0000 52.1504 2.9840 100.0000 72 46.8509 −7.1869 100.0000 52.4201 1.9729 100.0000 73 47.2193 −7.8548 100.0000 52.6845 0.9604 100.0000 74 47.5845 −8.5246 100.0000 52.9439 −0.0534 100.0000 75 47.9465 −9.1960 100.0000 53.1983 −1.0685 100.0000 76 48.3055 −9.8690 100.0000 53.4479 −2.0848 100.0000 77 48.6618 −10.5435 100.0000 53.6929 −3.1022 100.0000 78 49.0155 −11.2194 100.0000 53.9332 −4.1207 100.0000 79 49.3670 −11.8964 100.0000 54.1691 −5.1402 100.0000 80 49.7163 −12.5745 100.0000 54.4007 −6.1607 100.0000 81 50.0638 −13.2536 100.0000 54.6281 −7.1822 100.0000 82 50.4097 −13.9335 100.0000 54.8514 −8.2045 100.0000 83 50.7541 −14.6141 100.0000 55.0710 −9.2277 100.0000 84 51.0973 −15.2953 100.0000 55.2869 −10.2517 100.0000 85 51.4395 −15.9771 100.0000 55.4994 −11.2764 100.0000 86 51.7808 −16.6593 100.0000 55.7089 −12.3016 100.0000 87 52.1215 −17.3418 100.0000 55.9156 −13.3275 100.0000 88 52.4616 −18.0246 100.0000 56.1196 −14.3539 100.0000 89 52.8013 −18.7076 100.0000 56.3213 −15.3807 100.0000 90 53.1409 −19.3907 100.0000 56.5203 −16.4081 100.0000 91 53.4804 −20.0737 100.0000 56.7166 −17.4360 100.0000 92 53.8200 −20.7568 100.0000 56.9094 −18.4645 100.0000 93 54.1599 −21.4397 100.0000 57.0981 −19.4939 100.0000 94 54.5002 −22.1224 100.0000 57.2814 −20.5242 100.0000 95 54.8412 −22.8047 100.0000 57.4584 −21.5556 100.0000 96 55.1830 −23.4867 100.0000 57.6287 −22.5881 100.0000 97 55.5258 −24.1681 100.0000 57.7925 −23.6216 100.0000 98 55.8699 −24.8489 100.0000 57.8558 −24.6566 100.0000 99 56.4085 −25.3585 100.0000 57.1555 −25.3786 100.0000 100 57.1555 −25.3786 100.0000

It will also be appreciated that the airfoil 52 disclosed in the above Table I may be scaled up or down geometrically for use in other similar turbine designs. Consequently, the coordinate values set forth in Table I may be scaled upwardly or downwardly such that the airfoil profile shape remains unchanged. A scaled version of the coordinates in Table I would be represented by X, Y and Z coordinate values of Table I, with the X, Y and Z non-dimensional coordinate values converted to inches, multiplied or divided by a constant number.

An important term in this disclosure is “profile”. The profile is the range of the variation between measured points on an airfoil surface and the ideal position listed in Table I. The actual profile on a manufactured turbine rotor blade will be different than those in Table I and the design is robust to this variation meaning that mechanical and aerodynamic function are not impaired. As noted above, a + or −5% profile tolerance is used herein. The X, Y and Z values are all non-dimensionalized relative to the airfoil height.

The disclosed airfoil shape optimizes and is specific to the machine conditions and specifications. It provides a unique profile to achieve 1) interaction between other stages in the turbine 10; 2) aerodynamic efficiency; and 3) normalized aerodynamic and mechanical blade loadings. The disclosed loci of points defined in Table I allow the gas turbine 12 or any other suitable turbine to run in an efficient, safe and smooth manner. As also noted, any scale of the disclosed airfoil 52 may be adopted as long as 1) interaction between other stages in the pressure turbine 10; 2) aerodynamic efficiency; and 3) normalized aerodynamic and mechanical blade loadings are maintained in the scaled turbine.

This written description uses examples to disclose the invention, including the best mode, and also to enable any person skilled in the art to practice the invention, including making and using any devices or systems and performing any incorporated methods. The patentable scope of the invention is defined by the claims, and may include other examples that occur to those skilled in the art. Such other examples are intended to be within the scope of the claims if they have structural elements that do not differ from the literal language of the claims, or if they include equivalent structural elements with insubstantial differences from the literal languages of the claims. 

What is claimed is:
 1. A turbine rotor blade including an airfoil having an airfoil shape, the airfoil having a nominal profile in accordance with Cartesian coordinate values of X, and Z set forth in Table I, wherein the Cartesian coordinate values of X, Y and Z are non-dimensional values from 0% to 100% convertible to dimensional distances in inches by multiplying the Cartesian coordinate values of X, Y and Z by a height of the airfoil in inches, and wherein X and Y are distances in inches which, when connected by smooth continuing arcs, define airfoil profile sections at each distance Z, the airfoil profile sections at Z distances being joined smoothly with one another to form a complete airfoil shape, wherein the airfoil shape lies in an envelope within +/−5% of a respective coordinate value in a direction normal to any airfoil surface location.
 2. The turbine rotor blade as in claim 1, forming part of a stage of a turbine.
 3. The turbine rotor blade as in claim 1, wherein the turbine rotor blade forms a part of a first stage of turbine rotor blades of a turbine of a gas turbine.
 4. The turbine rotor blade as in claim 1, wherein the height of the turbine rotor blade is 5 inches to 20 inches.
 5. A turbine rotor blade including an airfoil having a suction-side uncoated nominal airfoil profile in accordance with suction-side Cartesian coordinate values of X, Y and Z set forth in Table I, wherein the Cartesian coordinate values of X, Y and Z are non-dimensional values from 0% to 100% convertible to dimensional distances in inches by multiplying the Cartesian coordinate values of X, Y and Z by a height of the airfoil in inches, and wherein X and Y are distances in inches which, when connected by smooth continuing arcs, define airfoil profile sections at each Z distance, the airfoil profile sections at the Z distances being joined smoothly with one another to form a complete suction-side airfoil shape, the X, Y and Z distances being scalable as a function of the same constant or number to provide a scaled-up or scaled-down airfoil, wherein the suction-side airfoil shape lies in an envelope within +/−5% of a respective coordinate value in a direction normal to any suction-side airfoil surface location.
 6. The turbine rotor blade as in claim 5, forming part of a stage of a turbine.
 7. The turbine rotor blade as in claim 5, wherein the turbine rotor blade forms a part of a first stage of turbine rotor blades of a turbine of a gas turbine.
 8. The turbine rotor blade as in claim 5, wherein the height of the turbine rotor blade is about 5 inches to about 20 inches.
 9. A turbine comprising a turbine wheel having a plurality of turbine rotor blades, each of the turbine rotor blades including an airfoil having a suction-side airfoil shape, the airfoil having a nominal profile in accordance with suction-side Cartesian coordinate values of X, Y and Z set forth in Table wherein the Cartesian coordinate values of X, Y and Z are non-dimensional values from 0% to 100% convertible to dimensional distances by multiplying the Cartesian coordinate values of X, Y and Z by a height of the airfoil, and wherein X and Y are distances which, when connected by smooth continuing arcs, define airfoil profile sections at each Z distance, the airfoil profile sections at the Z distances being joined smoothly with one another to form a complete suction-side airfoil shape, wherein the airfoil shape lies in an envelope within +/−5% of a respective coordinate value in a direction normal to any airfoil surface location.
 10. The turbine as in claim 9, wherein the turbine rotor blade is a first stage turbine rotor blade.
 11. The turbine as in claim 9, wherein X represents a distance parallel to the turbine axis of rotation.
 12. The turbine as in claim 9, wherein the height of the turbine rotor blade is about 5 inches to about 20 inches.
 13. The turbine as in claim 9, wherein each of the turbine rotor blades includes an airfoil having a pressure-side airfoil shape, the airfoil having a nominal profile in accordance with pressure-side Cartesian coordinate values of X, Y and Z set forth in Table I wherein the pressure-side Cartesian coordinate values of X, Y and Z are non-dimensional values from 0% to 100% convertible to dimensional distances by multiplying the pressure-side Cartesian coordinate values of X, Y and Z by the height of the airfoil, and wherein X and Y are distances which, when connected by smooth continuing arcs, define airfoil profile sections at each Z distance, the airfoil profile sections at Z distances being joined smoothly with one another to form a complete airfoil shape.
 14. The turbine as in claim 13, wherein the turbine rotor blade is a first stage turbine rotor blade.
 15. The turbine as in claim 9, wherein X represents a distance parallel to the turbine axis of rotation.
 16. The turbine as in claim 13, wherein the height of the turbine rotor blade is about 5 inches to about 20 inches. 